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Density Funtional Theory • Hohenberg-Kohn theorems – Energy of the system is a unique functional of the
charge density – The correct ground state charge density minimizes the
energy functional and the resulting energy is the ground state energy.
• Exchange-Correlation functional:unknown – Approximations: LDA, GGA (PW91, PBE)
• Kohn-Sham scheme – Map interacting system to non-interacting system
｠Well-convergedを calculation • Choose XC functional
– Accuracy: LDA < GGA < Hybrid
• k-points in Brillouin zone – Do k-point convergence
• Accuracy increases with no. of k-points – For metals
• Choose a smearing technique – Tetrahedron method – good for very accurate total
energy calculations. – Gaussian, Methfessel-Paxton etc. – good for ionic
and geometric relaxations
• Energy cutoff – Controls completeness of planewave basis
set
• Pseudopotentials – Replaces nucleus and core electrons with
fixed effective potential
What is VASP? • One of the software packages that uses DFT to solve the
quantum problem for materials – Uses periodic boundary conditions – Uses pseudopotential method with a plane waves basis set – Can model systems with maximum no. of atoms in the range of 100-200 – Commercial software package ~$5000 – Manual http://cms.mpi.univie.ac.at/vasp/vasp/PREC_tag.html
• Other DFT packages – (public license) – Quantum espresso (www.quantum-espresso.org) – Abinit( www.abinit.org)
– Siesta (http://icmab.cat/leem/siesta/)
VASP input files • INCAR – User specified parameters that define the calculation
• Global break condition, Energy cutoff, Smearing, Ionic and geometric relaxation parameters
• POSCAR – Specifies the periodic simulation cell – Information regarding geometry of the system
• POTCAR – Pseudopotential (PP) file – Information on PP and XC functional
• KPOINTS – Defines k-point mesh
Our system
• Al • 4 atoms, fcc structure – Translational lattice vectors • a1 = (4.0; 0.0; 0.0) • a2 = (0.0; 4.0; 0.0) • a3 = (0.0; 0.0; 4.0)
– Basis vectors • 0.0; 0.0; 0.0 • 0.5; 0.0; 0.5 • 0.5; 0.5; 0.0 • 0.0; 0.5; 0.5
POTCAR file with LDA XC
• For all elements VASP provides PP files in different flavors – US-PP, PAW-LDA, PAW-GGA, PAW-PBE
• If multiple species of atoms are in the system – Concatenate the same type of POTCAR files to make a single POTCAR file
• E.g. cat POTCAR_Al POTCAR_Mg >> POTCAR – The order of the POTCAR types should correspond to the ordering of atoms
in POSCAR
POSCAR file
• In 】Direct‒ coordinates the positions of atoms are given as a fraction of translational lattice vectors. – In 】Cartesian‒ coordinates position of atoms is given in global coordinates.
• Universal scaling factor scales the volume of the unit cell – If negative value is given then it is taken as the volume of the unit cell.
• Order of atom types in POSCAR should be same with POTCAR
INCAR file
• Minimal setup
– PREC : Controls the overall precision of the calculation
• Default: Normal – Standard calculation
– are sufficiently accurate with errors less than 1 meV/atom
• Low – quick and dirty calculations
• Accurate – High accuracy calculations
– ENCUT : Controls the no. of plane waves in the basis set
• Default: ENCUT=Largest ENMAX of POTCAR file
– Best if convergence test could be done
• For geometric relaxations (changing cell size) set ENCUT=1.3*ENMAXlargest
– LREAL: use of reciprocal/real space
• Default: .False. – uses in reciprocal space
• AUTO – recommended for systems more than 20 atoms. – uses real space
– ISMEAR
• Default: 1 – Methfessel-Paxton
• -5: Tetrahedron method
KPOINTS file • For Al bulk system For Al single atom
• k- point grid should be inversely propotional to the ratio of lattice vectors (else it will give rise to errors) – E.g. If the lattice vectors are a1=3 a2=9 a3=21 then k-point grid
should be 7:3:1 ratio
Comments on k-points and smearing
• Smearing – Guidelines for ISMEAR
• For semiconductors or insulators always use tetrahedron method (ISMEAR=-5) or if cell is too large use Gaussian smearing (ISMEAR=0)
• For relaxations in metals always use ISMEAR=1 – Never use ISMEAR > 0 for insulators and semiconductors.
• For very accurate total energy calculations (no relaxations in metals) use the tetrahedron method (ISMEAR=-5).
• k- point convergence – recommended to use even meshes (e.g. 8 × 8 × 8 ) for up to � = 8
• After that odd meshes are more efficient (e.g. 11 × 11 × 11)
– http://cms.mpi.univie.ac.at/vasp/vasp/Automatic_k_mesh_generation.html – http://cms.mpi.univie.ac.at/vasp/vasp/Number_k_points_method_smearing.html
#im-smear
Standard output of a VASP run: PREC=N
• N – iteration count • E – Total energy • dE – change of total
energy
• ncg – no. of evaluations of the Hamiltonian acting on WF
• rms – approximate error in WF • rms(c) – difference between
input and output charge density
Standard output of a VASP run: PREC=A
• No wrap-around errors • Energy difference ~ 0.00001eV = 0.01meV • Therefore PREC=Normal with EDIFF=1e-04 is accurate enough. • Data from stdout are written to OSZICAR file as well. • http://cms.mpi.univie.ac.at/vasp/vasp/stdout_OSZICAR_file.html
OUTPUT files
• CONTCAR – Contains position of the system after the calculation
has completed
• OSZICAR – Contains data of electronic steps and ionic steps
• OUTCAR – Complete output of run including input file data.
• CHGCAR – Charge density of system after run
Bulk Relaxations: Setup • ISIF tag
– determines which degrees of freedom (ions, cell volume, cell shape) are allowed to change.
– To only optimize ionic positions with in the cell do ISIF=2
– To perform cell volume and cell shape optimization do ISIF=2
• NSW tag: Specifies the maximum no. of ionic steps – Default NSW = 0
• IBRION tag: determines how the ions are
updated and moved. – If away from minimum use IBRION = 2
(Conjugate -gradient) – If close to minimum use IBRION=1 (quasi-
newton) – http://cms.mpi.univie.ac.at/vasp/vasp/IBRI
ON_tag_NFREE_tag.html
• Since Al is metallic – Use ISMEAR = 1 for
relaxation run • ENCUT is increased to
1.3*ENMAX to accommodate cell volume change.
Bulk relaxation:Procedure 1. Using the relaxation setup INCAR do the first run
with ISMEAR=1 (since metallic) and ISIF=2/3
2. Next copy CONTCAR to POSCAR, clear output files (use ‘cleanvaspfiles’) and run again – For ISIF=2 second step is unnecessary
3. Final step: switch on tetrahedron method
(ISMEAR=-5) to get very accurate energies – NEVER use energies from a relaxation run if volume
change is allowed.
• http://cms.mpi.univie.ac.at/vasp/vasp/Accurate_bulk_relaxations_with_internal_parameters_one.html
Volumetric relaxation procedure
• Best method is to use do ionic relaxations (ISIF=2) at different volumes and plot energy vs volume curve. – Fit to equation of state and obtain equilibrium volume
and energy
– ‘evfit’program offers several equation of state equations which can be used for fitting purposes.
• At the end of relaxation (ISIF =2/3) – if converged to parameters specified stdout will say
Automated Energy-Volume curve • Shell script • Uses lattice parameters from 3.5 Å
to 4.5 Å ~ only 10 points – More points will increase
accuracy • Extracts
– Final volume/ion – Energy/atom – Time for calculation
• Makes 2 output files (2 columns) – Energy/ion for each lattice
parameter – Energy/ion for each volume/ion
• Calculation at each fixed volume could be static or only ionic relaxations.
Curve fitting
• Using ‘evfit’ program we fit the data to – Murnaghan’s equation of state
• From the fitting we derive – Equilibrium lattice constant, bulk
modulus, and energy per atom.
Automated k-point convergence test
• EV generation script is called for different k-point grids – 4x4x4, 8x8x8 and 11x11x11
• At each fixed volume static calculation is done
• Total cpu time for these grids is recorded
• From 4x4x4 grid to 11x11x11 grid the time taken quadruples
• Energy difference is – From 4x4x4 to 8x8x8 ~50 meV – Form 8x8x8 to 11x11x1 ~ 6 meV
pbs script • To submit jobs to TALON or
RAPTOR you need a pbs command script – Make sure that total no. of
processors you request is used in your mpirun command • mpirun –np <nodes x ppn>
• When doing calculations over 20 atoms use cluster
For RAPTOR
For TALON
Surface energy calculations • First to very accurate bulk energy calculation
– Do k-point convergence to ~< 1meV • ｠if you want to calculate surface energies within ｱｰ meV you should probably
increase the k-mesh till your energy is converged to 1 meVを -VASP – Do Volumetric and ionic optimizations to a high degree
(PREC=Accurate, EDIFF=1e-06)
• Next Setup surface calculation – Start with a system that has at least 5 layers of atoms and 5 layers of
vacuum. • Good rule of thumb: make system big enough that surface to surface distance is
at least 10 Å – applies for any defect (vacancy, interstitial, stacking fault)
– Do k-point convergence test • Since in one direction there is a vacuum and it is longer than other translational
lattice vectors set revelent k-point to 1 • For the other 2 directions start off with values used for bulk calculation and do
convergence
• Finally keeping the k-points and all other parameters fixed change no. of bulk and vacuum layers – Look for convergence of surface energy – Remember to divide by 2 since periodic boundary conditions give rise
to 2 surfaces
Most common mistakes • Considering energy difference from calculations
that has different energy-cutoffs – If ENCUT is not specified default is largest ENMAX of
POTCAR – If adsorption energy is considered • E.g. adsorption of O atom on Al (111), Al ENMAX = 240 eV
while Co ENMAX = 400 eV
• Considering energy difference from calculations with two different k-point grids
• Please read – http://cms.mpi.univie.ac.at/vasp/vasp/most_important
_parameters_source_errors.html
How to validate results • Increase cutoff – Recommended to set ENCUT by
hand – Change to PREC=Accurate
• Increase no. of k-points • Increase slab thickness for
surface energy and adsorption energy calculations
• Increase the size of the supercell for defects – To remove interaction of defect
with defect due to periodic boundary condition
Homework 1 • Each group will use different XC functional
– LDA, GGA-PW 91 and GGA-PBE • DFT calculations
– Energy-volume curves – Elastic constants – Vacancy formation energy – Interstitial formation energy (Both octahedral and tetrahedral) – Surface formation energy for (111), (110) and (001) surfaces – Extrinsic and intrinsic stacking fault energies – Plot generalized stacking fault curve
• List of properties of Aluminum going up to atomistic scale – Lattice constant, bulk modulus, elastic constants – Defect energies – vacancy formation and interstitial – Stacking fault energies – Intrinsic and extrinsic – Surface formation energies – E-V plot and GSF plot
• Divide calculations within group – Report on ICME website what/how you did calculations assigned to you. – Explain whether you got a 】well-converged‒ result – You will submit a report and do evaluations of your group members
Visulization tools • VESTA – http://jp-minerals.org/vesta/en/ – Windows, Mac and Linux
• VMD – http://www.ks.uiuc.edu/Research/vmd/ – Windows, Mac and Linux
• OVITO – www.ovito.org/ – Windows and Linux
• XCrysDen – www.xcrysden.org/ – Mac and Linux
References
• VASP documentation – http://cms.mpi.univie.ac.at/vasp/vasp/
• VASP workshop slides – http://cms.mpi.univie.ac.at/vasp-
workshop/slides/documentation.htm
• VASP tutorial @ ugent – http://users.ugent.be/~devpouck/dlstuff/VAPS_tutorial
11_01_2010.pdf
